Hypothesis: HR 52:
Hahn-Banach Theorem: If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall x \in S)( f(x) \le p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\).
Conclusion: HR 288-n:
If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N2(n)\) A generalization of \(\cal N2\) | This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\) |
| \(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\) | \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\) |
Code: 3
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